In mathematics, an antiunitary transformation, is a bijective antilinear map

between two complex Hilbert spaces such that

for all and in , where the horizontal bar represents the complex conjugate. If additionally one has then U is called an antiunitary operator.

Antiunitary operators are important in quantum theory because they are used to represent certain symmetries, such as time-reversal symmetry. Their fundamental importance in quantum physics is further demonstrated by Wigner's theorem.

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Invariance transformationsEdit

In quantum mechanics, the invariance transformations of complex Hilbert space   leave the absolute value of scalar product invariant:

 

for all   and   in  .

Due to Wigner's theorem these transformations fall into two categories, they can be unitary or antiunitary.

Geometric InterpretationEdit

Congruences of the plane form two distinct classes. The first conserves the orientation and is generated by translations and rotations. The second does not conserve the orientation and is obtained from the first class by applying a reflection. On the complex plane these two classes corresponds (up to translation) to unitaries and antiunitaries, respectively.

PropertiesEdit

  •   holds for all elements   of the Hilbert space and an antiunitary  .
  • When   is antiunitary then   is unitary. This follows from
     
  • For unitary operator   the operator  , where   is complex conjugate operator, is antiunitary. The reverse is also true, for antiunitary   the operator   is unitary.
  • For antiunitary   the definition of the adjoint operator   is changed into[original research?]
     .
  • The adjoint of an antiunitary   is also antiunitary and
      (This is not to be confused with the definition of unitary operators, as the antiunitary operator   is not complex linear.)[original research?]

ExamplesEdit

  • The complex conjugate operator     is an antiunitary operator on the complex plane.
  • The operator
     
    where   is the second Pauli matrix and   is the complex conjugate operator, is antiunitary. It satisfies  .

Decomposition of an antiunitary operator into a direct sum of elementary Wigner antiunitariesEdit

An antiunitary operator on a finite-dimensional space may be decomposed as a direct sum of elementary Wigner antiunitaries  ,  . The operator   is just simple complex conjugation on  

 

For  , the operator   acts on two-dimensional complex Hilbert space. It is defined by

 

Note that for  

 

so such   may not be further decomposed into  's, which square to the identity map.

Note that the above decomposition of antiunitary operators contrasts with the spectral decomposition of unitary operators. In particular, a unitary operator on a complex Hilbert space may be decomposed into a direct sum of unitaries acting on 1-dimensional complex spaces (eigenspaces), but an antiunitary operator may only be decomposed into a direct sum of elementary operators on 1- and 2-dimensional complex spaces.

ReferencesEdit

  • Wigner, E. "Normal Form of Antiunitary Operators", Journal of Mathematical Physics Vol 1, no 5, 1960, pp. 409–412
  • Wigner, E. "Phenomenological Distinction between Unitary and Antiunitary Symmetry Operators", Journal of Mathematical Physics Vol1, no5, 1960, pp.414–416

See alsoEdit